Evaluate a Function Defined in Terms of an ODE

Evaluate a Function Defined in Terms of an ODE
Syntax

# include <cppad/speed/ode_evaluate.hpp>

ode_evaluate(x, m, fm)

Purpose
This routine evaluates a function

f : R^{n} \to R^{n}

defined by

f (x) = y (x, 1)

where

y (x, t)

solves the ordinary differential equation

\begin{matrix} y (x, 0) & = & b (x) \\ \partial_{t} y (x, t) & = & g [x, y (x, t), t] \end{matrix}

where

b : R^{n} \to R^{n}

and

g : R^{n} \times R^{n} \times R \to R^{n}

are not any further specified. A numerical method is used to solve the ode and obtain an accurate approximation for

y (x, 1)

. This in turn is used to compute values and Jacobian of the function

f (x)

.

Inclusion
The template function ode_evaluate is defined in the CppAD namespace by including the file cppad/speed/ode_evaluate.hpp (relative to the CppAD distribution directory). It is only intended for example and testing purposes, so it is not automatically included by cppad.hpp .

Float
The type Float must be a NumericType . The Float operation sequence for this routine will depend on the value of the argument x and hence it must be retaped for each new value of

x

.

x
The argument x has prototype



     const CppAD::vector<Float> &x

It contains he argument value for which the function, or its derivative, is being evaluated. The value

n

is determined by the size of the vector x .

m
The argument m has prototype



     size_t m

It is either zero or one and specifies the order of the derivative of

f (x)

, with respect to

x

, that is being evaluated.

m = 0
In this case the function

f (x)

is evaluated as described above.

m = 1
In this case the following extended system is solved:

\begin{matrix} y (x, 0) & = & b (x) \\ \partial_{t} y (x, t) & = & g [x, y (x, t), t] \\ y_{x} (x, 0) & = & b^{(1)} (x) \\ {partial}_{t} y_{x} (x, t) & = & \partial_{x} g [x, y (x, t), t] + \partial_{y} g [x, y (x, t), t] y_{x} \end{matrix}

fm
The argument fm has prototype



     CppAD::vector<Float> &fm

The input value of the elements of fm does not matter.

Function
If m is zero, fm has size equal to

n

and contains the value of

y (x, 1)

.

Gradient
If m is one, fm has size equal to n^2 and for

i = 0, \dots and n -1

j = 0, \dots, n -1

\frac{\partial y [i]}{\partial x [j]} (x, 1) = fm [i \cdot n + j]

Example
The file ode_evaluate.cpp contains an example and test of ode_evaluate.hpp. It returns true if it succeeds and false otherwise.

Source Code
The file ode_evaluate.hpp contains the source code for this template function.

Input File: cppad/speed/ode_evaluate.hpp